OFFSET
1,3
COMMENTS
How many times does each prime appear in this sequence?
The only value (prime(n) - k!) = 0 is at n=1, where k=2.
Are n=2, k=2 and n=4, k=3 the only occurrences of (prime(n) - k!) = 1?
There exist infinitely many solutions of the form (prime(n) - k!) = prime(n-t), t < n.
Are there infinitely many solutions of the form (prime(n) - k!) = prime(r_1)*...*prime(r_i); r_i < n for all i?
From Bernard Schott, Jul 16 2021: (Start)
Answer to the second question is no: 18 other occurrences (n,k) of (prime(n) - k!) = 1 are known today; indeed, every k > 1 in A002981 that satisfies k! + 1 is prime gives an occurrence, but only a third pair (n, k) is known exactly; and this comes for n = 2428957, k = 11 because (prime(2428957) - 11!) = 1.
The next occurrence corresponds to k = 27 and n = X where prime(X) = 1+27! = 10888869450418352160768000001 but index X is not yet available (see A062701).
LINKS
Jinyuan Wang, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = prime(n)- k! where k is the greatest number for which k! <= prime(n).
a(n) = A212598(prime(n)). - Michel Marcus, Feb 19 2019
EXAMPLE
a(1) = prime(1) - 2! = 2 - 2 = 0;
a(2) = prime(2) - 2! = 3 - 2 = 1;
a(3) = prime(3) - 2! = 5 - 2 = 3;
a(4) = prime(4) - 3! = 7 - 6 = 1;
a(5) = prime(5) - 3! = 11 - 6 = 5;
a(6) = prime(6) - 3! = 13 - 6 = 7;
a(7) = prime(7) - 3! = 17 - 6 = 11;
a(8) = prime(8) - 3! = 19 - 6 = 13;
a(9) = prime(9) - 3! = 23 - 6 = 17;
a(10) = prime(10) - 4! = 29 - 24 = 5.
MAPLE
f:=proc(n) local p, i; p:=ithprime(n); for i from 0 to p do if i! > p then break; fi; od; p-(i-1)!; end;
[seq(f(n), n=1..70)]; # N. J. A. Sloane, May 22 2012
PROG
(PARI) a(n) = my(k=1, p=prime(n)); while (k! <= p, k++); p - (k-1)!; \\ Michel Marcus, Feb 19 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Ctibor O. Zizka, Apr 02 2008
EXTENSIONS
More terms from Jinyuan Wang, Feb 18 2019
STATUS
approved